Strong colorings of hypergraphs
Geir Agnarsson1 and Magnús M. Halldórsson2
1
Department of Mathematical Sciences, George Mason University,
MS 3F2, 4400 University Drive, Fairfax, VA 22030,
[email protected]
2
Department of Computer Science, University of Iceland,
Dunhaga 3, IS-107 Rvk, Iceland,
[email protected]
Abstract. A strong vertex coloring of a hypergraph assigns distinct colors to vertices that are contained in a common hyperedge. This captures
many previously studied graph coloring problems. We present nearly
tight upper and lower bound on approximating general hypergraphs,
both offline and online. We then consider various parameters that make
coloring easier, and give a unified treatment. In particular, we give an
algebraic scheme using integer programming to color graphs of bounded
composition-width.
Keywords: hypergraph, strong coloring, approximation, composition
width.
1
Introduction
The purpose of this article is to discuss the properties of a special kind of vertex
coloring of hypergraphs, where we insist on vertices that are contained in a
common hyperedge receiving distinct colors. Such strong colorings capture a
number of graph coloring problems that have been treated separately before.
A hypergraph H = (V, E) consists of a finite set V = V (H) of vertices and a
collection E = E(H) ⊆ P(V ) of subsets of V . A strong coloring of H is a map
Ψ : V (H) → N such that whenever u, v ∈ e for some e ∈ E(H), we have that
Ψ (u) 6= Ψ (v). The corresponding strong chromatic number χs (H) is the least
number of colors for which H has a proper strong coloring.
Strong coloring can be viewed as a regular vertex coloring problem of the
clique graph Gc (H) (also known as 2-section graph or representing graph [6]) of
the hypergraph H, defined on the same set of vertices, with edge set E(Gc (H)) =
{{u, v} : u, v ∈ e for some e ∈ E(H)}. In this way, χs (H) = χ(Gc (H)), the
ordinary chromatic number of the clique graph.
We consider both online and offline coloring algorithms. We analyze them
in terms of their competitiveness or approximation factor, respectively, which
in both cases is the maximum ratio between the number of colors used by the
algorithm on an instance to the chromatic number of the instance. In the standard online graph coloring problem, the graph is presented one vertex at a time
along with edges only to the previous vertices. Each time the algorithm receives
a vertex, it must make an irrevocable decision as to its color.
Since the hypergraph H is our original input, we would like to use all the
associated parameters that come with it, and use those to either obtain an
optimal strong coloring or a good approximation. This is relevant because it is
often difficult to deduce the hypergraph from the graph representation, or to
find the ”best” such hypergraph. Finding the smallest clique hypergraph H that
is equivalent to a graph G, i.e. such that Gc (H) = G, is equivalent to finding
the smallest Clique Cover (GT 17: Covering by Cliques in [13]), which is hard
to approximate within n2− factor, for any > 0 [12].
One of the main objectives of this research is in opening a new line of research
by unifying several coloring problems as strong coloring appropriate types of hypergraphs. We have gathered a host of results on these problems by modifying
and sometimes slightly extending previous results on graph coloring. Finally, we
have made the first step into a systematic treatment of parameters of hypergraphs and their clique graphs that make solution or approximation easier.
1.1
Instances of strong colorings problems
Down-coloring DAGs Our original motivation to study strong hypergraph
colorings stems from a digraph coloring problem that occurs when bounding
storage space in genetic databases. A down-coloring of a DAG (acyclic digraph)
G is coloring of the vertices so that vertices that share a common ancestor receive different colors. One motivation for such a coloring (see [2]) is to provide
an efficient structure for querying relational tables referencing the digraph, including the retrieval of rows in a given table that are conditioned based on sets
of ancestors from G.
For a DAG G, the binary relation ≤ on V (G) defined by u ≤ v ⇔ u = v,
or there is a directed path from v to u in G, is reflexive, antisymmetric, and
transitive, and is therefore a partial order on V (G). We denote by max{G} the
set of source vertices of G, i.e., the maximal vertices with respect to this partial
order ≤. For vertices u, v ∈ V (G) with u ≤ v, we say that u is a descendant
of v. The down-set D[u] of a vertex u ∈ V (G) is the set of all descendants
of u in G, that is, D[u] = {x ∈ V (G) : x ≤ u}. As defined in [3], the downhypergraph HG of a DAG G contains the same set of vertices with the downsets E(HG ) = {D[u] : u ∈ max{G}} of sources of G as hyperedges. A downcoloring of a digraph corresponds to a strong coloring of the corresponding downhypergraph.
Note that not all hypergraphs are down-hypergraphs of a DAG, but they are
easily recognized: We say that a hypergraph has the unique element property if
each hyperedge contains a vertex not contained in any other hyperedge. We can
observe that a hypergraph H is a down-hypergraph of some DAG iff H has the
unique element property [3].
Further properties between graphs, hypergraphs and the posets yielding them
can be found in [30], where the corresponding clique graph associated with the
poset is called an upper bound graph of the poset.
Distance-2 coloring graphs A distance-2 coloring of a graph is a vertex
coloring where vertices at distance two or less must receive different colors. This
problem has received attention for applications in frequency allocation [37, 26],
where two stations must use a different frequency if they are both to be able to
communicate with a common neighbor. Another application given in [29] relates
to the partition of the columns of a matrix for parallel solution so that columns
solved in the same iteration do not share a non-zero element in the same row.
The neighborhood hypergraph NG of a graph G consists of the same vertex
set with a hyperedge consisting of the closed neighborhood N [v] = {u : u =
v or {u, v} ∈ E(G)} of each vertex v ∈ V (G). A strong coloring of NG is equivalent to a distance-2 coloring of G.
The distance-2 coloring problem is also equivalent to an ordinary coloring
problem on the square graph G2 of the graph G. The k-th power Gk of a graph
G is a graph on the same vertex set, with an edge between any pair of vertices
of distance at most k in G. The square graph is indeed the clique graph of the
neighborhood hypergraph. While it is easy to compute the power graph Gk from
G, Motwani and Sudan [34] showed that it is NP-hard to compute the k-th root
G of a graph Gk , for any k ≥ 2. On the other hand, it is not hard to deduce the
original graph when given its neighborhood hypergraph.
McCormick [29] was the first to show that the problem of coloring the power
of a graph√ is NP-complete, for any fixed power. He gave a greedy algorithm
with a O( n)-approximation for squares of general graphs, which was matched
by the NP-hardness of an Ω(n1/2− )-approximation, for any > 0 [4]. Several
recent papers have studied distance-2 coloring planar graphs [26, 5], for which
the current best upper bound is 1.66∆(G) + O(1) colors due to Molloy and
Salavatipour [33].
1.2
Related coloring results
The best current upper bound for approximating ordinary graph coloring is
O(n(lg lg n/ lg n)3 ) [15], while it is hard to approximate within n1− factor, for
any > 0 [7]. For a survey on graph coloring approximations, see [36].
The weak hypergraph coloring problem is an alternative generalization of the
graph coloring problem, where the vertices are to be colored so that no hyperedge is monochromatic. Several results are known about such approximations,
including a Ω(n1− ) hardness [25].
Each color class in a strong coloring is called a strong independent set (strong
stable set). The k-set packing problem is equivalent to the strong independent
set problem in degree-k hypergraphs, by looking at the dual graph. This is NPhard to approximate within factor O(k/ lg k) [20]. This suggests, but does not
guarantee, that coloring degree-k hypergraphs is hard to do within an asymptotic
factor much smaller than k.
Strong coloring a hypergraph H is also equivalent to edge coloring the dual
hypergraph H ∗ . Kahn [23] showed that χ0 (H) ≤ ∆ + o(∆), if no two hyperedges
share many vertices. Further improvements were obtained by Molloy and Reed
[32].
1.3
Overview of paper
In the following section, we introduce the general version of the problem, which
involves multicolorings, where a set of colors is to be assigned to each vertex.
This follows naturally from some preprocessing of the hypergraph instance. In
Section 3, we consider several parameters of graphs and hypergraphs and analyze
their effect on approximability. In Section 4, we give bounds for online and offline
strong coloring algorithms on general graphs. Finally, in Section 5, we present
the technically most involved part of the paper, with a polynomial time coloring
algorithm for the class of k-composite graphs.
2
Hypergraph contractions and multicolorings
To describe the strong coloring problem in its full generality, we must introduce
multicolorings.
Multicoloring For a simple graph G let ν : V (G) → N be a natural weight. By
a multicoloring of (G, ν), we mean an assignment c̃ : V (G) → P(N) to the power
set of N, such that (i) |c̃(u)| = ν(u) for each u ∈ V (G), and (ii) {u, v} ∈ E(G) ⇒
c̃(u) ∩ c̃(v) = ∅. The corresponding multichromatic number χ̃(G, ν) is then the
smallest k which allows a legitimate multicoloring c̃ : V (G) → P({1, . . . , k}).
In general, we may thus be given a weighted hypergraph, for which we seek
a strong multicoloring. This corresponds to a multicoloring of the clique graph,
whose weight function is identical to its corresponding hypergraph.
Contractions One reason why it may be natural to generalize the problem
to multicolorings is to handle certain contractions, or operations that simplify
the instance. We consider particularly contractions that involve vertices with
identical neighborhoods.
A hypermodule is a set S of vertices that appear identical to vertices outside S,
i.e., for u ∈ V \ S and v, w ∈ S, then {u, v} ∈ E(Gc (H)) iff {u, w} ∈ E(Gc (H)).
A contraction takes a weighted hypergraph (H, ν) and a hypermodule S and
produces a smaller reduced hypergraph (H 0 , ν 0 ) where S has been replaced by
a single vertex of weight χ(Gc (H[S]), ν). One now can show that χs (H, ν) =
χs (H 0 , ν 0 ).
Furthermore, degrees in the reduced hypergraph are no greater than before:
for any vertex v we have dH 0 (v) ≤ dH (v). Thus, any result for approximation
or time complexity involving degrees, number of vertices, or number of edges,
carries over for the reduced hypergraph.
We may want to limit the kind of contractions that we seek. In particular,
within our context, it is natural to search for clique contractions, where the clique
graph Gc (H[S]) induced by S is a clique. In this case, degrees remain unchanged.
Note that a set of vertices that are contained in exactly the same hyperedges
of H is a hypermodule that induces a clique in Gc (H). By viewing each such
hypermodule S of H as a single vertex uS , and connecting two such vertices if,
and only if, they are both contained in a common hyperedge of H, we obtain the
reduced graph Gr (H). This reduced graph Gr (H) further has a natural weight
ν : V (Gr (H)) → N given by ν(uS ) = |S|. Hence, H yields a corresponding
weighted reduced graph (Gr (H), | · |), something we will use in Section 5.
Hypergraph contraction preserves both chordality and perfectness. A polynomial time algorithm for multicoloring perfect graphs was given by Grötschel,
Lovász, and Schrijver [8], under the problem name of weighted coloring.
3
Parameters of graphs and hypergraphs
The largest cardinality of a hyperedge of H will be denoted by σ(H) and the
largest cardinality of a hypermodule corresponding to a vertex of Gr (H) will be
denoted by µ(H). Clearly we have µ(H) ≤ σ(H).
Maximum degree For a vertex u ∈ V (H) of a hypergraph H, its degree dH (u)
is the number of hyperedges that contain the vertex u. Note that the degree of u
is usually much smaller than the number of neighbors of u (that is, the number
of vertices contained in a common hyperedge with u.) The minimum (maximum)
degree of a vertex in H is denoted by δ(H) (∆(H)).
Hypergraphs of degree at most t have the property that their clique graphs
are (t + 1)-claw free, i.e. contain no induced star on (t + 2)-vertices. This ensures
that almost any coloring obtains a ratio of at most t, even online.
Call an online coloring algorithm frugal if it does not introduce a new color
unless it is forced to do so, i.e. if the corresponding vertex is already adjacent to
vertices of all other colors. The First-Fit algorithm is clearly frugal. An offline
algorithm is frugal if each vertex assigned a color i is adjacent to vertices of each
color 1, . . . , i − 1.
Lemma 1. Any frugal coloring algorithm is at most ∆(H)-competitive for the
clique graph of a hypergraph H.
Clique graphs of degree-2 hypergraphs contain the class of line graphs, and
thus strong coloring such hypergraphs subsumes the edge coloring problem of
multigraphs. This is hard to approximate within an absolute ratio of less than
4/3, but can be done using at most 1.1χ0 (G) + 0.7 colors [35], where χ0 is the
edge chromatic number.
We can obtain an incomparable bound in terms of the maximum degree of
the clique graph, by extending an approach of [14] to multicolorings.
Theorem 1. Multicoloring can be approximated within d(∆(G) + 1)/3e.
This uses the following specialization of a lemma of Lovász [27]. It can be implemented in linear time [14] by first assigning the vertices greedily in order
to the color class to which they have the fewest neighbors, followed by local
improvement steps that move a vertex to a class with fewer neighbors.
Lemma 2. Let G be a graph and let t = d(∆ + 2)/3e. There is a partition of
V (G) into sets V1 , . . . , Vt such that the graph G[Vi ] induced by each Vi is of
maximum degree at most 2.
Given the graphs of maximum degree 2 promised by the lemma, we can color
each of them optimally in linear time. (Details omitted.) Thus, we obtain a
t-approximation to the multicoloring problem.
For unweighted graphs, a better approximation bound of d(∆ + 1)/4e can be
obtained [14], by using that graphs of maximum degree 3 can be colored optimally by way of Brooks theorem. It, however, does not apply for multicolorings.
Inductiveness By the inductiveness (or the degeneracy) of H, denoted by
ind(H), we mean the parameter defined by ind(H) = maxS⊆V (H) {δ(H[S])}.
Here, H[S], for a vertex subset S denotes the subhypergraph of H induced by S,
or the hypergraph with edge set E(H[S]) = {X ∩S : X ∈ E(H) and |X ∩S| ≥ 2}.
Recall that the inductiveness naturally relates to a greedy coloring of a graph
G that uses at most ind(G) + 1 colors (see [5]). The degree of a vertex v in
Gc (H) is at most (σ(H) − 1) times its degree in H. Thus, the observation of
[3] that ind(Gc (H)) ≤ ind(H)(σ(H) − 1). Hence, we have an ind(H)-factor
approximation by the greedy algorithm. For ordinary graphs, we can obtain a
simple contraction in the approximation by a factor of nearly 2. We observe here
that this holds also for multicolorings. The following observation is from [16].
Theorem 2. Suppose we are given a graph G with vertex weights w and a Ccoloring of G (i.e., a partition of the vertex set into independent sets). Then we
can approximate the multichromatic number within a factor of dC/2e.
Corollary 1. Multicoloring can be approximated within d(ind(G) + 1)/2e.
The inductiveness measure is useful for bounding the performance of online
algorithms. Irani [21] showed that the First-Fit coloring algorithm uses at most
O(ind(G) lg n) to color an n-vertex graph G.
Corollary 2. The First-Fit coloring algorithm is O(ind(H) lg n)-competitive for
coloring the clique graph of a hypergraph H.
Composition width First we define a modular decomposition [22], which is
also called substitution decomposition as in [31], and has been studied widely,
since it is by many considered one of three most important hierarchical graph
decomposition, the others being tree decomposition [38] and the graph decomposition upon which clique-width is defined [11].
Definition 1. Let G be a graph with V (G) = {u1 , . . . , uk } If G1 , . . . , Gk are
graphs, then let G0 = GhG1 , . . . , Gk i denote the graph obtained by replacing each
vertex ui in G by the graph Gi , and connect each vertex in Gi to each vertex in
Gj if, and only if, ui and uj are connected in G0 . In this case we say that G
is a modular decomposition of G1 , . . . , Gk . The induced subgraphs Gi of G0 are
called modules of G0 .
Definition 2. We call a graph G0 k-composite if it is a null graph, or recursively, if there is a graph G on ` ≤ k vertices and k-composite graphs G1 , . . . , G` ,
such that G0 = GhG1 , . . . , G` i. The composition-width of G0 , denoted cow(G0 ),
is the least k for which G0 is k-composite.
Remarks: (i) Every null graph is 1-composite and every clique is 2-composite.
(ii) If G is k-composite, then G is k 0 -composite for each k 0 ≥ k. (iii) Every graph
on n vertices is n-composite.
Clique-width By a labeled graph G we mean a graph G provided with a labeling
function ι : V (G) → N. Consider the following four graph operations, introduced
in [11]: (i) Create a new vertex u with a label i. (ii) Form the disjoint union of
labeled graphs G1 and G2 , denoted by G1 ⊕G2 . (iii) Connect all i-labeled vertices
with all the j-labeled vertices, where i 6= j (iv) Relabel all vertices labeled i with
the label j, where i 6= j. The clique-width of a graph G, denoted by cw(G) is the
least number of labels needed so that G can be constructed by the above four
graph operations.
Observation 3 For a graph G we have cw(G) ≤ cow(G). If further cw(G) ∈
{1, 2} then we have cw(G) = cow(G).
Clique-width was first defined and studied in [11] and it generalizes the notion of
treewidth, introduced in [38], that is, a graph of bounded treewidth is necessarily
also of bounded clique-width [10], but not conversely since a clique of arbitrary
size has clique-width of two, while its treewidth is the number of vertices in the
clique.
Problems definable in a certain variations of Monadic Second Order Logic,
including maximum independent set, are solvable in polynomial time for graphs
of bounded clique-width [9]. However, graph coloring has been shown to be
not one of those (see [9]). Still, it has been shown that for a fixed k ∈ N, the
chromatic number of a graph G of clique-width of at most k, can be determined
2k+1
+1
in time O(23k+1 k 2 n2
) [24]. This however depends on that the expression
that forms the graph using the four operations above is given.
4
Approximations for down-colorings
In this section we give
√ bounds on the approximability of strong coloring general
hypergraphs. The O( m)-approximation of strong independent sets of [18] leads
to an equivalent approximation of strong coloring. However, we here obtain a
bound on the inductiveness in terms of an arbitrary number m of edges, and
obtain an approximation in terms of σ(H), the largest hyperedge size.
√
Theorem 4. For a hypergraph H with m edges, ind(Gc (H)) ≤ mσ(H).
√
Proof. Let k = m. Let S be a vertex subset inducing a subgraph of Gc (H)
of minimum degree ind(Gc (H)). Let H[S] be the subgraph of H induced by S,
and let mS be the number of hyperedges in it. If there is a vertex of degree
at most k in H[S], then its degree in Gc (H[S]) (which is at least ind(Gc (H))
is at most kσ(H), in which case the theorem follows. Otherwise, each vertex
is of degree at least k and the number of edge-vertex incidences is at least
k|S|. It follows that the average edge size in H[S] is at least k|S|/mS , and thus
σ(H) ≥ k|S|/mS ≥ k|S|/m = |S|/k.
Now, since the inductiveness of Gc (H) is equal to the minimum degree of
Gc (H[S]) which has at most |S| vertices, we have that ind(Gc (H[S])) ≤ |S|−1 <
σ(H) · k and the theorem follows.
Corollary 3. There is a greedy algorithm
that approximates
the strong coloring
√
√
of hypergraphs within a factor of m. This yields a M approximation for the
down-coloring of DAGs, where M = | max{G}| is the number of source vertices
in G.
Observe that we bounded the number of colors used by the algorithm in terms
of the maximum edge size σ(H). Thus, we have shown that the strongpchromatic
number
of a hypergraph H differs from σ(H) by a factor of at most |E(H)| =
√
m. In terms of down-graphs and hypergraphs, we obtain the following bound.
q
√
Corollary 4. χ(Gc (HG )) = χs (HG ) ≤ | max{G}| · σ(HG ) ≤ n · σ(HG ).
Compare the above corollary with [3, Obs. 2, p. 306], which shows that the
bounds given can be obtained.
4.1
Approximation hardness
We now give a reduction from the ordinary coloring problem that shows that the
approximation of the greedy algorithm is close to best possible. The Ω(n1/2− )hardness result for distance-2 coloring of [4] already yields the same hardness for
strong coloring hypergraphs with n vertices and n edges. We show here similar
result for down-hypergraphs and restricted types of DAGs.
Given a graph G0 , we construct a DAG G of height two by letting V (G) =
E(G0 ) ∪ V (G0 ) and E(G) = {(e, v) : v ∈ e, v ∈ V (G0 ), e ∈ E(G0 )}. The digraph
has a source vertex for each edge in G0 , a leaf vertex for each node in G0 , and
an edge from a source to a leaf if the leaf corresponds to a vertex incident on
the edge corresponding to the source vertex.
Let H = HG be the corresponding down-hypergraph. Note that the subhypergraph H[V (G0 )] induced by the leaves is a graph and is exactly the graph
G0 . The source nodes of H induce an independent set. Thus, χ(G0 ) ≤ χs (H) ≤
χ(G0 ) + 1. In fact, χs (H) = χ(G0 ) if χ(G0 ) ≥ 3. By the results of Feige and
Kilian [7], the chromatic number problem cannot be approximated within a
factor of |V (G)|1− , for any > 0, unless NP ⊆ ZPP, i.e. unless there exist
polynomial-time
p randomized algorithms for NP-hard problems. Here, we have
|V (G0 )| = Ω( |V (H)|) and hence the following.
Theorem 5. It is hard to approximate the down-coloring of DAGs within a
factor of n1/2− , for any > 0. This holds even for digraphs of height two.
We may now ask if it is possible to give a better approximation for important
special cases of the down-coloring problem. In particular, digraphs arising from
pedigrees (i.e. records of ancestry for people) have some special properties; in
particular, each vertex has in-degree at most 2, and normally a fairly small
out-degree. We can show that even in this case, we cannot do better. (Proof
omitted.)
Theorem 6. It is hard to approximate the down-coloring of DAGs within a
factor of n1/2− , for any > 0, even when restricted to DAGs of in-degree and
out-degree two.
4.2
Online coloring
In the standard online graph coloring problem, the graph is presented one vertex
at a time along with edges only to the previous vertices. Each time the algorithm
receives a vertex, it must make an irrevocable decision as to its color [19, 17].
The hypergraph model might lead to a different model, e.g. where all the
vertices contained in edges incident on previous vertices are given. It can be
inferred from the arguments below that this does not produce a great advantage.
Applying the result of Irani cited earlier, we obtain the following upper bound
for online coloring.
√
Corollary 5. First-Fit is O( m lg n)-competitive for the clique graph√of a hypergraph H with m edges and n vertices. More generally, it is O( m lg n)competitive for graphs that can be covered with m cliques.
In the case of distance-2 coloring, one can argue a better bound. Namely,
let G be the underlying graph and HG be its neighborhood hypergraph. Then,
∆(H) = ∆(G) + 1 and σ(H) = ∆(G). Thus, the competitive ratio
√ of any any
frugal online coloring algorithm is at most min(∆(H), n/σ(H)) ≤ n.
√
Proposition 1. Any frugal online coloring algorithm is n-competitive for the
distance-2 coloring graphs.
On the hardness side, lower bounds for online graph coloring carry over for
clique graphs, simply by viewing the graphs as hypergraphs. Halldórsson and
Szegedy [19] showed that for any online algorithm, there is a lg n-colorable graph
on n vertices for which the algorithm uses Ω(n/ lg n) colors. This holds also for
randomized algorithms against an oblivious adversary. This was later extended
to a known graph model, where a graph isomorphic to the (fixed) input graph
is given in advance [17]. Thus, there is one particular graph that is hard for
any online coloring algorithm. By padding this graphs with isolated vertices (or
small cliques), we can have this hold for graphs of any density.
Lemma 3. For any n and m, there is a particular graph with n vertices
√ and at
most m cliques such that any online coloring algorithm is at least Ω( m/ lg2 m)
competitive.
5
Multicoloring, an algebraic approach
In this section we will consider an algebraic approach to determine the strong
chromatic number of a given hypergraph by using integer programming. We
then show how the same method can recursively yield an improved poly-time
algorithm to obtain an optimal coloring for k-composite graphs.
Observation 7 For a hypergraph H and its weighted reduced graph (Gr (H), |·|),
we have χs (H) = χ̃(Gr (H), | · |).
Here we view a given hypergraph H as weighted graph (Gr (H), | · |), since by
Observation 7 we have χs (H) = χ̃(Gr (H), | · |). Hence, we will here consider
multicolorings of a weighted graph (G, ν), where ν : V (G) → N is a natural
weight.
Consider our weighted graph (G, ν) where V (G) = {u1 , . . . , uk }. Let G0 =
GhQ1 , . . . , Qk i be the modular decomposition of the cliques Q1 , . . . , Qk , where
each Qi has ν(ui ) vertices. Clearly we have that χ̃(G, ν) = χ(GhQ1 , . . . , Qk i),
and so the computation of χ̃(G, ν) can be trivially reduced to the computation
of the chromatic number of a graph G0 . However, taking further into the account
the structure of GhQ1 , . . . , Qk i, we can shorten the computations considerably,
especially when the ν(ui )’s are large compared to k. We proceed as follows:
For each proper (i.e. nonempty) independent set I ⊆ V (G) we form a variable
xI . We denote by I(G) the set of all independent sets of G, and for each u ∈ V (G)
we denote by I(G; u) ⊆ I(G) the set of all independent sets of G that contain
the vertex u. For each u ∈ V (G) we form the following constraint
X
xI = ν(u).
(1)
I∈I(G;u)
Let us fix a listing of the elements of I(G): For an ordering V (G) = {u1 , . . . , uk },
note that each U = {ui1 , . . . , ui` } yields a word U 7→ word(U ) = ui1 · · · ui` , and
hence the sets of V (G) can be ordered lexicographically, viewing u1 , . . . , uk as an
ordered alphabet. We now can list the elements of P(V (G)) degree lexicographically, or by deglex in short, in the following way [1]:
|Ul | < |U2 | or
U1 < U2 ⇔
(2)
|Ul | = |U2 | and word(U1 ) < word(U2 ) lexicographically.
With the deglex ordering (2), we can form the |I(G)|-tuple x of the variables xI ,
and the constraints from (1), determined by the k vertices of G, can be written
collectively as A(G)·x = n, where n = (ν(u1 ), . . . , ν(uk )) and A(G) is a uniquely
determined |I(G)| × k matrix with only 0 or 1 as entries. Note that the sum Σ
of all the variables xI can be given by the dot-product Σ = 1 · x, where 1 is the
|I(G)|-tuple with 1 in each of its entry.
Theorem 8. For an integer weighted graph (G, ν) the multichromatic number
χ̃(G, ν) is given by the integer program
χ̃(G, ν) = min{1 · x : A(G) · x = n, x ∈ (N ∪ {0})|I(G)| },
(3)
where A(G) is uniquely determined by (1) and (2), and n = (ν(u1 ), . . . , ν(uk )).
It is well-known that the problem of solving an integer programming problem P
as (3) is NP-complete. However, considering the complexity in terms of
n = u∈V (G) ν(u) (which corresponds to the number of vertices in the original
hypergraph H) and assuming that k = |V (G)| is fixed and “small” compared to
n, it is worthwhile to discuss complexity analysis.
Lemma 4. A connected graph G on k vertices has at most 2k−1 proper independent sets.
By Lemma 4 the number of variables xI in x is at most 2k−1 . Note also that
by our deglex ordering, A(G) is already in reduced row echelon form with its
first k × k submatrix being the k × k identity matrix Ik . If N = maxu∈V (G) ν(u),
then clearly each optimal solution x must satisfy 0 ≤ xI ≤ N for each I ∈ I(G).
Hence, we have at most 2k−1 − k free variables xI in x, namely those xI with
|I| ≥ 2, each taking value in {0, 1, . . . , N }. We now make some rudimentary
computational observations, which are asymptotically tight for general G:
To check whether a set of i vertices is independent or not, we need 2i edge
Pk
k i
comparisons. Hence, I(G) can be obtained by at most
i=1 i 2 = k(k −
k−3
1)2
comparisons. Having I(G), to determine I(G; u) for each u ∈ V (G)
we need at most k2k−1 operations. In all, determining the linear program (3)
we need at most k(k − 1)2k−3 + k 2 2k−1 < k 2 2k operations. Further, for each
given value of x, the expressions A(G) · x and 1 · x can be evaluated in at most
k2k−1 + 2k−1 < k2k steps. Hence, we have the following.
Observation 9 For a connected integer weighted graph (G, ν) with |V (G)| = k
and N = maxu∈V (G) ν(u), an optimal χ̃(G, ν)-multicoloring can be obtained by
k−1
k−1
k2k (k + (N + 1)2 −k ) operations, or in O(k2k (N + 1)2 −k ) time for k fixed.
With the notation from previous Section 2 we therefore have the following for a
hypergraph.
Corollary 6. For a hypergraph H with its reduced graph Gr (H) on k vertices, the complexity of obtaining an optimal strong χs (H)-coloring is given by
k−1
O(k2k (µ(H) + 1)2 −k ).
Note that our complexities are polynomial expressions, only because we assume
k to be fixed here.
We conclude this section by considering the complexity of obtaining an optimal coloring of a k-composite graph by using the integer program (3) recursively.
The following is clear.
Proposition 2. Let k ∈ N be given. The chromatic number χ(G0 ) of a kcomposite graph G0 = GhG1 , . . . , G` i where ` ≤ k, is given by χ(G0 ) = χ̃(G, ν),
where the integer weight ν : V (G) → N is given by ν(ui ) = χ(Gi ) for each
i ∈ {1, . . . , `}.
By Observation 9 we can obtain a χ(G)-coloring by at most k2k (k + (N +
k−1
k−1
1)2 −k ) operations, or in O(k2k (N + 1)2 −k ) time, where N = maxi χ(Gi ),
provided that we have an optimal coloring for each of the Gi ’s. If that is not the
case however, we proceed recursively, but with more care, since we need at this
point to keep track of the actual upper bound of arithmetic operations when
applying the recursion. For a rooted tree (T, r) let Tu be the subtree rooted at
u ∈ V (T ).
Lemma 5. Let (T, r) be a rooted tree with n leaves, where each internal node
has at least two children. If f : N → R is positive and non-decreasing, then
X
u∈V (T )
f (|V (Tu )|) ≤ Sf (n) := (n − 1)f (1) +
n
X
f (i).
i=1
Remark: The bound of Sf (n) from Lemma 5 is tight: Consider the degenerate
binary tree T on 2n − 1 vertices and n leaves, where each
Pinternal vertex has a
leaf as a left child. In this case we have for any f that u∈V (T ) f (|V (Tu )|) =
Pn
(n − 1)f (1) + i=1 f (i) = Sf (n).
An upper bound of the number of arithmetic operations need to obtain an
optimal χ(G0 )-coloring of a k-composite graph G0 , can
P now be obtained from the
weighted rooted module tree (TG0 , | · |) of G0 by u∈V (TG0 ) f (w(uG00 )), where
each vertex uG00 corresponds to an induced subgraph (strong module) G00 of
G0 , and w(uG00 ) = |V (G00 )|, and the function f from Observation 9 is f (n) =
Pk
k−1
k2k (k + (n + 1)2 −k ), since N = max1≤i≤k χ(Gi ) ≤ i=1 |V (Gi )| = n.
Theorem 10. Provided the modular decompositions defining the k-composite
connected graph G on n vertices are given, the number of arithmetic operations
k−1
k−1
needed to obtain an optimal χ(G)-coloring is k22 +1 n + 4k(n + 2)2 −k+1 and
k−1
hence can be obtained in O(kn2 −k+1 ) time, for k fixed.
As mention above, the chromatic number of a connected graph G on n vertices of
2k+1
+1
clique-width at most k, can be computed in O(23k+1 n2
) time [24], provided
that the corresponding k-expression for G is given. By Observation 3 that also
holds for k-composite graphs as well. However, the bound given in Theorem 10
is considerably better than the mentioned bound from [24]. In addition, it is not
known, for k ≥ 4, whether there exists a polynomial time algorithm to obtain a
k-expression for a graph G of clique-width at most k.
For a graph G0 known to have a module decomposition, a strong modular
decomposition (where the modules do not have a nonempty intersection) which is
unique, can be computed in O(n2 ) time, since such a decomposition of a graph is
a special case of a modular decomposition a 2-structure (a slightly more general
concept than a graph) [22]. However, in such a strong module decomposition
G0 = GhG1 , . . . , G` i it could be that ` > k. But, if it is known that G0 is kcomposite, then the union of some of the strong modules G1 , . . . , G` will make
a single module of G0 . To check this is the same as to check for modules in G,
which will take at most O(n2 ) time as well. Hence, the module decomposition
of G0 constituting a k-decomposition will take at most O(n2 ) time at each step,
which is much less than f (n) from above, right before Theorem 10. Therefore,
unlike for graphs of clique-width of k or less, we have the following corollary of
Theorem 10, where we do not assume the modular k-decomposition.
Corollary 7. The complexity of obtaining an optimal coloring for a k-composite
k−1
graph on n vertices is O(kn2 −k+1 ).
Note: A few words of warning are in order. If we do not have the modular
decompositions, then we do need to compute the k-decomposition at each step
of the recursive definition of a k-composite graph. A priori it could look as if the
constant hidden in the O(n2 ) term might affect the overall complexity. However,
a rooted tree with n leaves and no degree-2 internal vertex has at most 2n − 1
vertices. So, the overall computation is at most (2n − 1)O(n2 ) = O(n3 ) which is
dominated by the expression in Theorem 10.
Acknowledgments
The authors are grateful to Walter D. Morris Jr. for many helpful discussions
regarding integer programming.
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